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Basic properties of regular-closed functions

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Summary

A functionf∶X→Y is defined to be regular-closed if for each regular-closedA⊂X, f(A) is closed inY. Numerous theorems are presented which give properties of such functions as well as sufficient conditions for a function to be regular-closed. Comparisons are also made between regular-closed functions and certain other types of non-continuous functions. A sample of the theorems proved in this paper would be as follows:

Theorem. A functionf∶X→Y is regular closed if and only if for each open or regularclosedA⊂X, cl(f(A))⊂f(cl(A)).

Theorem. Normality is preserved under continuous regular-closed surjections.

Theorem. Letf∶X→Y be a function such that the induced graph function is almostcontinuous and almost-open. LetX beH-closed andY Hausdorff. Thenf is regular-closed.

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Authors and Affiliations

  1. Dept. of Math., University of Arkansas, Fayettville, Arkansas, U. S. A.

    Paul E. Long & Larry L. Herrington

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  1. Paul E. Long
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  2. Larry L. Herrington
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Long, P.E., Herrington, L.L. Basic properties of regular-closed functions. Rend. Circ. Mat. Palermo 27, 20–28 (1978). https://doi.org/10.1007/BF02843863

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  • DOI: https://doi.org/10.1007/BF02843863

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